Equipment designed to fit in bore holes is, by nature, long and slim. Torsional loading of the shafts is often a limiting factor in the maximum power developed. Shaft connections are critical weak points in the shaft designs. The preferred choice for shaft connection in an Electrical Submersible Pump (ESP) is a spline, either a multi tooth involute, or six tooth square spline. Both have proved to be quick and reliable for field assembly of the equipment. Shaft material strength requirements for high power units can exceed 200,000 PSI yield. This is reaching the limits of readily available materials. To achieve these strength levels, elongation (ductility) has to be sacrificed. The selected materials are therefore more brittle and more susceptible to fatigue in the areas of high stress concentration.
The object of this invention is to reduce the high concentration of stress by making it possible to more uniformly distribute the forces involved along the spline. The problem of the spline joint is, when under load, the coupling which is stiffer will twist much less than the shaft. This is due to the difference in diameter. The angle of twist per unit length in a shaft varies directly as the applied torque, inversely as the modulus of shear and the forth power of the diameter. The coupling, even though it is a hollow cylinder, has to be extremely thin to match the angle of twist of the shaft. This is impractical because the coupling thickness is necessary to counter the radial forces that are attempting to expand (explode) the coupling. For example, if the shaft is 1″ diameter and the coupling is 1.0″ ID and 1.5″ OD, the coupling is five times stiffer than the shaft. Under the same torque load, it will only have ⅕ of the twist as a shaft of the same length.
Examining failed shafts shows that almost the entire load is distributed over a short area where the end of the coupling engages the shaft spline. The length of this “work area” can only expand as the twist of the coupling allows more contact with the shaft and the local mashing (yielding) of the spline tooth itself. The solution to this situation in this invention is to shape the spline so that the initial contact area begins at the end of the shaft in the middle of the coupling. As the shaft twists in relation to the coupling, the contact area increases, the stress is spread over an increasing area, reducing the stress concentration at the initial contact location.
The coupling is formed by a pull broach, and it would be an extremely difficult operation to create a coupling spline that was wider at the start (mouth) of the coupling and taper as it progressed inward. The shaft spline is more easily modified. For a square tooth spline on which the sides of each tooth are parallel, the cutter would have to progress down the shaft with a slight twist angle to the axis of the shaft. It would also be necessary that the mating coupling spline tooth be cut wide enough to accommodate this twisted spline. A simpler arrangement could be made for the involute. These splines are generally made on a hobbing machine. The shaft is rotated and a rotating cutter, timed to cut the spline teeth progresses along the shaft from the end to the maximum design length of the spline. If the cutter plunges into the shaft material as it progress along the spline it will result in a tooth width that is wider at the end of the shaft and tapers as it moves into the spline. This will force the coupling to make first contact in the depth of the coupling and the contact area increases in the direction of the mouth of the coupling as the shaft material twists in relation to the coupling. The plunging cutter method will not work on a true square tooth form, but can be applied if the form is modified so that the shaft tooth tapers towards the OD of the shaft.
The present invention addresses stress concentrations that occur in a splined shaft that is involved with the transfer of torque with a coupling. This concentration is caused by the comparative rigidity of the female spline coupling which encompasses the male spline shaft. This difference in magnitude is sufficient to allow an assumption in the design that the female spline is rigid and all the torsional elastic displacement is concentrated in the shaft in a very narrow band of material at the intersection of the mouth of the coupling and shaft. As the mechanical limits of the equipment are pushed, stronger shaft materials are the option of choice. These metals obtain strength at the sacrifice of ductility and a heightened sensitivity to crack propagation reduction in fatigue resistance.
To lessen the burden on the shaft it is necessary to increase the area of the torsional contact beyond the narrow band. Increasing the compliance of the coupling by reducing its diameter is impractical because it increases the radial stress in the coupling from the radial component of the torque developed by the spline tooth. The coupling could fail from expansion (explosion). The possible solution is to have the axial tooth contact line designed so that as the torsion deforms the shaft elastically, the contact line becomes longer. This can be accomplished by changing form of the involute or square tooth so that it slightly spirals in the opposite direction of the torque loading in anticipation of the a elastic strain that occurs with loading. This is just marginally practical. Basically a large amount of tedious work is required for the benefit achieved. A simpler method of the present invention is to sacrifice a very small amount of shaft strength. This method requires that tooth form be developed so the tooth cutting hob plunges slightly towards the center of the shaft as the cut runs from the end of the shaft to its maximum length. The result is a “tapered tooth” wider at the end of the shaft which will first make contact with the coupling near its center and extends its contact line as the shaft deformation increases.
The following US Patents deal with the issue of splined shafts and couplings and the general design considerations that are involved in these designs but do not touch on the issue of stress distribution in such designs as addressed by the present invention: U.S. Pat. Nos. 3,073,134; 4,133,516; 4,331,006; 6,056,511; 6,120,261; 6,126,416; 6,332,841 and 6,709,234.
Those skilled in the art will gain a better understanding of the invention from the description of the preferred embodiment and associated drawings that appear below with an understanding that the full scope of the invention is given by the appended claims.